Mirai Kuriyama is pictured at the beginning of the episode about to attempt suicide by jumping off the roof of a building when the other series' main character Akihito Kanbara arrives. This is the first seen that placed me into a state of awe and shock, one that gripped me instantly into the plot of the anime. The biggest shock (as shown in the image above) is when Mirai does a back flip and attempts to stab and kill Akihito. Akihito doesn't die and they leave the scene asking who each other are given the circumstances. Personally I like anime that give me the feels and kick me in the face to keep my attention, and so far this anime does all of that all in the first scene.
The first episode of Beyond the Boundary acts as a way in which to bring us into the human and youmu world. While this episode culminates with a battle between Mirai and a youmu (once Akihito has convinced her to actually fight it) we aren't given a full explanation as to what is going on, but we don't need one yet. The first episode of the series is explicitly designed to grab our attention and act as visual candy that will be lent an expository set of episodes to follow. One of the interesting vague plot points that we are let known, is the fact that the character and head of the literary club, Mitsuki Nase, does not want Akihito to get involved with Mirai. We can only ponder as to the full reasons, but she links it to Mirai's ancestry and secretive past.
In the end of the episode Mirai finally starts to fight the youmu. Akihito is there but he seems to only be there to lend moral support. Honestly, I really like how Mirai's klutzy sublime nature and Spirit World Warrior blood wielding powers mesh. Having these abilities allow Mirai to shine as a lovable character and Akihito's fetish over glasses and her glasses adds to some of the comic relief so we don't get too bogged down in details and the world and characters being created. This show is a must watch for any self-respecting anime fan and is sure to yield a snowballing fan base as other viewers start to take notice of its polished style and engaging story.